This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A176731 #52 Nov 17 2024 03:22:52
%S A176731 1,12,504,45360,7076160,1698278400,580811212800,268334780313600,
%T A176731 161000868188160000,121716656350248960000,113196490405731532800000,
%U A176731 127006462235230779801600000,169172607697327398695731200000,263909268007830741965340672000000,476620138022142319989405253632000000
%N A176731 Denominators of coefficients of a series, called g, related to Airy functions.
%C A176731 The numerators are always 1.
%C A176731 f(z) := Sum_{n>=0} (1/b(n)) * z^(3*n) with b(n) := A176730(n) and g(z) := Sum_{n>=0} (1/a(n)) * z^(3*n+1) build the two independent Airy functions Ai(z) = c(1)*f(z) - c(2)*g(z) and Bi(z) = sqrt(3) * (c(1)*f(z) + c(2)*g(z)) with c(1) := 1/(3^(2/3) * Gamma(2/3)), approximately 0.35502805388781723926, and c(2) := 1/(3^(1/3) * Gamma(1/3)), approximately 0.25881940379280679840.
%C A176731 If y := Sum_{n >= 0} x^(3*n+1)/a(n), then y'' = x*y. - _Michael Somos_, Jul 12 2019
%H A176731 G. C. Greubel, <a href="/A176731/b176731.txt">Table of n, a(n) for n = 0..200</a>
%H A176731 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 , 10.4.2 - 5. [alternative scanned copy].
%H A176731 Wolfdieter Lang, <a href="/A176730/a176730.txt">The first 20 terms of the f(z) and g(z) functions</a>.
%H A176731 NIST's Digital Library of Mathematical Functions, <a href="http://dlmf.nist.gov/9.4">Airy and Related Functions (Maclaurin Series)</a> by Frank W. J. Olver.
%F A176731 a(n) = denominator((3^n) * risefac(2/3, n)/(3*n + 1)!) with the rising factorials risefac(k, n) := Product_{j=0..(n-1)} (k+j) and risefac(k, 0) = 1.
%F A176731 From _Peter Bala_, Dec 17 2021: (Start)
%F A176731 a(n) = 3*n*(3*n + 1)*a(n-1) with a(0) = 1.
%F A176731 a(n) = (3*n + 2)!/(n!*3^n)*Sum_{k = 0..n} (-1)^k*binomial(n,k)/(3*k + 2).
%F A176731 a(n) = (1/2)*(3*n + 2)!/(n!*3^n)*hypergeom([-n, 2/3], [5/3], 1).
%F A176731 a(n) = (2*Pi*sqrt(3))/9 *(1/3^(n+1))*Gamma(3*n+4)/( (n+1)*Gamma(1/3)* Gamma(n + 5/3) ). (End)
%F A176731 a(n) = (9^n*n!*(n + 1/3)!)/(1/3)!. - _Peter Luschny_, Dec 20 2021
%e A176731 Rational g-coefficients: [1, 1/12, 1/504, 1/45360, 1/7076160, 1/1698278400, 1/580811212800, 1/268334780313600, ...].
%p A176731 a := proc (n) option remember; if n = 0 then 1 else 3*n*(3*n+1)*a(n-1) end if; end proc: seq(a(n), n = 0..20); # _Peter Bala_, Dec 17 2021
%t A176731 a[ n_] := If[ n < 0, 0, -1 / (3^(1/3) Gamma[ 1/3] SeriesCoefficient[ AiryAi[ x], {x, 0, 3 n + 1}])]; (* _Michael Somos_, Oct 14 2011 *)
%t A176731 a[ n_] := If[ n < 0, 0, (3 n + 1)! / Product[ k, {k, 2, 3 n + 1, 3}]]; (* _Michael Somos_, Oct 14 2011 *)
%o A176731 (PARI) {a(n) = if( n<0, 0, (3*n + 1)! / prod( k=0, n-1, 3*k + 2))}; /* _Michael Somos_, Oct 14 2011 */
%Y A176731 Cf. A176730.
%K A176731 nonn,frac,easy
%O A176731 0,2
%A A176731 _Wolfdieter Lang_, Jul 14 2010